To determine $f(x)$ if it is given as:

$f(x) = \frac{2x - 3}{x^2 - 4}$

Step 1: Simplify the Denominator

Notice that $x^2 - 4$ is a difference of squares:

$x^2 - 4 = (x - 2)(x + 2)$

Thus, the function becomes:

$f(x) = \frac{2x - 3}{(x - 2)(x + 2)}.$

Step 2: Confirm Domain Restrictions

The denominator $x^2 - 4 = 0$ when $x = 2$ or $x = -2$. At these points, the denominator is zero, and the function is undefined. Therefore, the domain of $f(x)$ is:

$x \in \mathbb{R} \ \text{except} \ x = 2 \ \text{and} \ x = -2.$

Step 3: General Behavior of $f(x)$

• $f(x)$ is a rational function (a ratio of polynomials).
• The numerator $2x - 3$ is linear, while the denominator $(x - 2)(x + 2)$ is quadratic.
• Vertical asymptotes occur where the denominator is zero: $x = 2$ and $x = -2$.
• As $x \to \infty$ or $x \to -\infty$, the degree of the denominator (2) is greater than the degree of the numerator (1), so $f(x) \to 0$. Thus, the horizontal asymptote is $y = 0$.

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Final Answer:

$f(x) = \frac{2x - 3}{(x - 2)(x + 2)}, \quad x \neq 2, \ x \neq -2.$

Let me know if you'd like:

1. Steps to find derivatives of $f(x)$.
2. The limits at $x = 2$ or $x = -2$.
3. Steps to graph the function $f(x)$.
4. Behavior analysis near asymptotes.
5. Explanation of how horizontal asymptotes work.

Tip: For rational functions, always simplify expressions and check where the function is undefined!